Technology Category
Search and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionReasoning under Uncertainty: Stochastic Optimization
On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns
🤖 AI Summary
This work addresses the problem of determining the optimal message size in private information retrieval (PIR) schemes under arbitrary collusion patterns. Focusing on capacity-achieving decomposable PIR schemes, the study characterizes their structural properties and establishes, for the first time, a general lower bound on message size for uniform decomposable schemes based on the hitting number of server subsets. By integrating tools from combinatorics, information theory, and set cover theory, the analysis examines the family of subsets induced by collusion patterns. Tight lower bounds are derived for canonical settings—including T-collusion, disjoint collusion sets, and cyclic T-consecutive collusion—with matching achievable schemes constructed for the latter two cases, thereby fully characterizing the optimal message size in these scenarios.
Technology Category
Application Category
📝 Abstract
A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings. In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size.
Problem
Research questions and friction points this paper is trying to address.
Private Information Retrieval
Message Size
Collusion Pattern
Capacity-Achieving Scheme
Optimal Message Size
Innovation
Methods, ideas, or system contributions that make the work stand out.
Private Information Retrieval (PIR)
arbitrary collusion patterns
optimal message size
hitting number
capacity-achieving schemes
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